Higher order Poincaré-Pontryagin functions and iterated path integrals. (English) Zbl 1104.34024

Consider a nonintegrable foliation \(df-\varepsilon(Pdx+Qdy)=0\), where \(\varepsilon\) is a small parameter and \(f,P,Q\) are polynomials in the two variables \(x,y\). Assume that the unperturbed system \(df=0\) has an annulus of periodic trajectories \(\{\gamma(t)\}\), where \(\gamma(t)\subset\{f=t\}\) for \(t\in (a,b)\). The related first return map has the form \[ {\mathcal P}(t,\varepsilon)=t+\varepsilon^k M_k(t) +\varepsilon^{k+1} M_{k+1}(t)+\ldots, \] where \(k\) is a natural number. The zeroes of the generating function \(M_k\) in \((a,b)\) correspond to the limit cycles produced by the period annulus after the perturbation.
Here, the author takes an analytic continuation of the generating function and establishes several facts about it. It is proved that \(M_k(t)\) is a linear combination of iterated path integrals along \(\gamma(t)\) of length at most \(k\) over certain rational one-forms. \(M_k\) is a function of moderate growth and its monodromy representation is finite-dimensional. Moreover, \(M_k\) satisfies a linear differential equation of Fuchs type and its monodromy group is contained in \(\text{SL}(n,\mathbb{Z})\), where \(n\) is the order of the equation. One has \(n\leq r^k\), where \(r=\dim H_1(f^{-1}(t_0),{\mathbb Z})\) and \(t_0\) is a typical value of \(f\).


34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
Full Text: DOI arXiv Numdam EuDML Link


[1] Arnold, V.I. - Geometrical methods in the theory of ordinary differential equations, Grundlehr. Math. Wiss. , vol. 250, Springer-Verlag , New York (1988). · Zbl 0648.34002
[2] Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N. - Singularities of Differentiable Maps, vols. 1 and 2, Monographs in mathematics , Birkhäuser, Boston, 1985 and 1988.
[3] Briskin, M. , Yomdin, Y. - Tangential Hilbert problem for Abel equation , preprint, 2003. · Zbl 1097.34025
[4] Brudnyi, A. - On the center problem for ordinary differential equation, arXiv:math 0301339 (2003).
[5] Chen, K.-T.- Algebras of iterated path integrals and fundamental groups, Trans. AMS156, p. 359-379 (1971). · Zbl 0217.47705
[6] Chen, K.-T. - Iterated Path Integrals, Bull. AMS83 (1977) 831-879. · Zbl 0389.58001
[7] Françoise, J.-P. - Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergod. Theory and Dyn. Syst.16, p. 87-96 (1996). · Zbl 0852.34008
[8] Françoise, J.-P. - Local bifurcations of limit cycles, Abel equations and Liénard systems, in Normal Forms, Bifurcations and Finitness Problems in Differential Equations, NATO Science Series II, vol. 137, 2004.
[9] Fulton, W. - Algebraic Topology, Springer , New York, 1995. · Zbl 0852.55001
[10] Gavrilov, L. - Petrov modules and zeros of Abelian integrals , Bull. Sci. Math.122, no. 8, p. 571-584 (1998). · Zbl 0964.32022
[11] Gavrilov, L. , Iliev, I.D. - The displacement map associated to polynomial unfoldings of planar vector fields, arXiv:math.DS/0305301 (2003). · Zbl 1093.34015
[12] Hain, R. - The geometry of the mixed Hodge structure on the fundamental group, Proc. of Simposia in Pure Math., 46, p. 247-282 (1987). · Zbl 0654.14006
[13] Hain, R. - Iterated integrals and algebraic cycles: examples and prospects. Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), p. 55-118, Nankai Tracts Math., 5, World Sci. Publishing, River Edge, NJ, 2002. · Zbl 1065.14012
[14] Hall, M. - The Theory of Groups, AMS Chelsea Publishing, 1976. · Zbl 0354.20001
[15] Hilbert, D. - Mathematische probleme, Gesammelte Abhandlungen III, Springer-Verlag, Berlin, p. 403-479 (1935).
[16] Ilyashenko, Y.S. - Selected topics in differential equations with real and complex time, in Normal Forms, Bifurcations and Finitness Problems in Differential Equations, NATO Science Series II , vol. 137, 2004, Kluwer. preprint, 2002.
[17] Jebrane, A., Mardesic, P., Pelletier, M. - A generalization of Françoise’s algorithm for calculating higher order Melnikov functions , Bull. Sci. Math.126, p. 705-732 (2002). · Zbl 1029.34081
[18] Jebrane, A., Mardesic, P., Pelletier, M. - A note on a generalization of Franoise’s algorithm for calculating higher order Melnikov function, Bull. Sci. Math.128, p. 749-760 (2004). · Zbl 1274.37012
[19] Parsin, A.N. - A generalization of the Jacobian variety , AMS Translations, Series 2, p. 187-196 (1969). · Zbl 0189.21501
[20] Passman, D. - The algebraic theory of group rings , Wiley, New York, 1977. · Zbl 0368.16003
[21] Pontryagin, L.S.- Über Autoschwingungssysteme, die den Hamiltonischen nahe liegen, Phys. Z. Sowjetunion6, 25-28 (1934) · Zbl 0010.02302
[22] ; On dynamics systems close to Hamiltonian systems, Zh. Eksp. Teor. Fiz.4, p. 234-238 (1934), in russian.
[23] Roussarie, R. - Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics , vol. 164, Birkhäuser Verlag , Basel (1998). · Zbl 0898.58039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.